<h2>Problem 156</h2>
<div style="color:#666;font-size:80%;">25 May 2007</div><br />
<div class="problem_content">
<p>Starting from zero the natural numbers are written down in base 10 like this:
<br />
0 1 2 3 4 5 6 7 8 9 10 11 12....
</p>
<p>Consider the digit <var>d</var>=1. After we write down each number <var>n</var>, we will update the number of ones that have occurred and call this number <var>f</var>(<var>n</var>,1). The first values for <var>f</var>(<var>n</var>,1), then, are as follows:</p>
<div style='text-align:center;'>
<table style='text-align:center;' align='center'>
<tr>
<td><var>n</var></td><td><var>f</var>(<var>n</var>,1)</td>
</tr><tr>
<td>0</td><td>0</td>
</tr><tr>
<td>1</td><td>1</td>
</tr><tr>
<td>2</td><td>1</td>
</tr><tr>
<td>3</td><td>1</td>
</tr><tr>
<td>4</td><td>1</td>
</tr><tr>
<td>5</td><td>1</td>
</tr><tr>
<td>6</td><td>1</td>
</tr><tr>
<td>7</td><td>1</td>
</tr><tr>
<td>8</td><td>1</td>
</tr><tr>
<td>9</td><td>1</td>
</tr><tr>
<td>10</td><td>2</td>
</tr><tr>
<td>11</td><td>4</td>
</tr><tr>
<td>12</td><td>5</td>
</tr></table></div>
<p>Note that <var>f</var>(<var>n</var>,1) never equals 3.
<br />
So the first two solutions of the equation <var>f</var>(<var>n</var>,1)=<var>n</var> are <var>n</var>=0 and <var>n</var>=1. The next solution is <var>n</var>=199981.</p>
<p>In the same manner the function <var>f</var>(<var>n,d</var>) gives the total number of digits <var>d</var> that have been written down after the number <var>n</var> has been written.
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In fact, for every digit <var>d</var> <img src='images/symbol_ne.gif' width='11' height='10' alt='&ne;' border='0' style='vertical-align:middle;' /> 0, 0 is the first solution of the equation <var>f</var>(<var>n,d</var>)=<var>n</var>.</p>
<p>Let <var>s</var>(<var>d</var>) be the sum of all the solutions for which <var>f</var>(<var>n,d</var>)=<var>n</var>.
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You are given that <var>s</var>(1)=22786974071.</p>
<p>Find  <img src='images/symbol_sum.gif' width='11' height='14' alt='&sum;' border='0' style='vertical-align:middle;' /> <var>s</var>(<var>d</var>) for 1 <img src='images/symbol_le.gif' width='10' height='12' alt='&le;' border='0' style='vertical-align:middle;' /> d <img src='images/symbol_le.gif' width='10' height='12' alt='&le;' border='0' style='vertical-align:middle;' /> 9.</p>
<p>Note: if, for some <var>n</var>, <var>f</var>(<var>n,d</var>)=<var>n</var>
 for more than one value of <var>d</var> this value of <var>n</var> is counted again for every value of <var>d</var> for which <var>f</var>(<var>n,d</var>)=<var>n</var>.</p>
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